PHYSICAL REVIEW B 66, 165217 共2002兲
Femtosecond pump-probe reflectivity study of silicon carrier dynamics
A. J. Sabbah and D. M. Riffe
Physics Department, Utah State University, Logan, Utah 84322-4415
共Received 28 May 2002; revised manuscript received 14 August 2002; published 29 October 2002兲
We have studied the ultrafast optical response of native-oxide terminated Si共001兲 with pump-probe reflectivity using 800 nm, 28 fs pulses at an excitation density of (5.5⫾0.3)⫻1018 cm⫺3 . Time-dependent reflectivity changes comprise third-order-response coherent-transient variations arising from anisotropic state filling
and linear-response variations arising from excited free carriers, state filling, and lattice heating. A time
constant of 32⫾5 fs associated with momentum relaxation is extracted from the coherent-transient variations.
The state-filling and free-carrier responses are sensitive to carrier temperature, allowing an electron-phonon
energy relaxation time of 260⫾30 fs to be measured. The recovery of the reflectivity signal back towards its
initial value is largely governed surface recombination: a surface recombination velocity of (3⫾1)
⫻104 cm s⫺1 is deduced for native-oxide terminated Si共001兲.
DOI: 10.1103/PhysRevB.66.165217
PACS number共s兲: 78.47.⫹p, 42.65.Re, 72.20.⫺i
I. INTRODUCTION
The continuing miniaturization of semiconductor devices
into the nanoscale regime has emphasized the importance of
understanding carrier dynamics on increasingly shorter time
scales. For example, in a nanoscale transistor electron transit
times can be a short as a few picoseconds. This time scale is
the domain of electron and phonon scattering, and in some
cases bulk and surface recombination processes. In order to
understand carrier behavior in ever smaller devices, sophisticated Monte Carlo1– 4 and hydrokinetic transport5 models
that incorporate both realistic band structure and carrierscattering processes have been developed. A significant result from the hydrokinetic transport theory is that the ratio of
carrier momentum-relaxation time to energy-relaxation time
has a significant impact on hot-carrier transport.5
Ultrafast laser based pump-probe techniques have been
instrumental in probing carrier dynamics in crystalline Si.
Four techniques have been particularly useful: transient reflectivity 共and/or transmission兲,6 –11 transient-grating diffraction 共including two-beam self-diffraction兲,12–15 photothermal
deflection,16,17 and time resolved photoemission.18 –23 In all
of these techniques a relatively strong laser pulse 共or two
crossed pulses in the case of transient grating diffraction兲
excites valence-band electrons into the conduction band. The
relaxation of these electrons and/or the redistribution of the
excess energy initially contained in these excited carriers is
then monitored with a time-resolved probe. From these studies the following picture of carrier dynamics in Si has
emerged. Initially, momentum-relaxing scattering processes
dephase the excited-carrier states on a time scale significantly shorter than 100 fs;15,24 theoretical calculations suggest that both carrier-carrier25 and carrier-phonon3,4,26 scattering contribute to the dephasing. Carrier-carrier scattering
also produces a well-defined electron temperature 共and presumably hole temperature兲 on a time scale of approximately
100 to 300 fs.21,23 Electron-phonon scattering also thermalizes the excited carriers with the lattice within a few hundred
fs.8,9,14 On longer time scales bulk7 and surface11 recombination processes operate to transfer the excited electrons back
to the valence band. At the same time that these local dy0163-1829/2002/66共16兲/165217共11兲/$20.00
namical processes take place, diffusion10,13 and/or bandbending induced transport of the excited carriers may
occur.18,19,21,23
Here we report on transient reflectivity measurements
from Si共001兲 using nearly Gaussian 28⫾1 fs 共FWHM兲
pulses from a Ti:sapphire oscillator operating at a center
wavelength ␭⫽800 nm (h ␯ ⫽E ph⫽1.55 eV). The small
pulse width combined with an excellent signal-to-noise ratio
provides a detailed look at the ultrafast dynamics of hot carriers in Si on time scales from the laser pulse width up to our
maximum delay time of 120 ps. From the data we obtain
quantitative information on carrier momentum relaxation,
electron-phonon energy relaxation, and surface mediated
electron-hole recombination. This ultrafast work determines
a momentum relaxation time in Si. Our measured electronphonon energy relaxation time is combined with earlier measurements of the same quantity at different excess carrier
energies.8,9,14 The combined experimental results compare
favorably with recent Monte Carlo calculations of holeenergy relaxation in Si.4 From the extracted surface recombination velocity we obtain an estimate for the surface defect
density of native-oxide terminated Si共001兲.
II. EXPERIMENTAL DETAILS
The ultrafast pump-probe reflectivity measurements were
performed with an in-house constructed Ti-sapphire oscillator, based on the design of Asaki et al.27 Interferometric autocorrelation traces obtained with a rotating-mirror autocorrelator are used to determine the laser-pulse widths and
confirm the nearly Gaussian nature of the pulses.28 Using a
half-wave plate and polarizing beam splitter the laser pulses
are separated into p- and s-polarized pump and probe pulses,
respectively. The normally incident 2.80⫾0.13 nJ pump
pulses are mechanically chopped and focused to a nearly
Gaussian shaped spot with a beam radius w 0 ⫽8.5 ␮ m and
on-axis fluence f c ⫽2.43⫾0.12 mJ cm⫺2 . The 33 times
weaker probe beam, incident at an angle of 45°, is elliptically shaped 共at the sample兲 with Gaussian beam radii w x
⫽6.2␮ m and w y ⫽4.3 ␮ m. The reflected probe pulses pass
through a linear polarizer, which suppresses scattered light
66 165217-1
©2002 The American Physical Society
PHYSICAL REVIEW B 66, 165217 共2002兲
A. J. SABBAH AND D. M. RIFFE
from the orthogonally polarized pump, and are detected with
a Si photoconductive detector. The phase-sensitive-detected
probe-beam signal 共divided by its dc component兲 is proportional to ⌬R/R (R is the reflectivity兲. Most of the runs were
obtained with the pump 关probe兴 beam polarization parallel to
the 共110兲 关 (11̄0) 兴 direction. However, in order to confirm
the nonlinear-response nature of the observed initial transient, several runs were also obtained with the sample rotated
about its azimuth so that the pump 关probe兴 beam polarization
was along the 共010兲 关共100兲兴 direction.
The initially excited carrier density in the near-surface
region can be calculated from
N 0 ⫽ 共 1⫺R n 兲 f eff
冉
冊
␣ ␤ 共 1⫺R n 兲 f eff
⫹
,
E ph 2E ph␶ FWHM
共1兲
where R n ⫽0.328 is the normal-incidence reflectivity,29
␶ FWHM⫽28 fs is the FWHM pulse width, f eff⫽ f c 兵 关 1
⫹(w x /w 0 ) 2 兴关 1⫹(w y /w 0 ) 2 兴 其 ⫺1/2⫽1.76⫾0.08 mJ cm⫺2 is
the effective incident pump fluence as seen by the probe, ␣
⫽1020 cm⫺1 is the linear absorption coefficient,29 and ␤
⫽6.8 cm GW⫺1 is the two-photon absorption-coefficient parameter. We have deduced this value of ␤ from previously
reported ultrafast measurements with 1.55 eV, 90 fs pulses
which showed that linear and two photon absorption are
equal at 40 mJ cm⫺2 fluence.14 Using Eq. 共1兲 we calculate
N 0 ⫽(5.5⫾0.3)⫻1018 cm⫺3 .
Measurements were made on a total of four Si共001兲
samples, both n and p type with RT carrier concentrations
between 1013 and 1015 cm⫺3 . The densities were chosen to
be well below the laser excited concentration N 0 in order to
simplify analysis of the reflectivity data. The RT carrier concentration of each sample was determined from the
resistivity,30 which was measured with a standard four-point
probe technique. Surface preparation prior to the resistivity
and reflectivity measurements consisted of dipping each
sample in concentrated HF acid for two minutes and then
rinsing in ultrapure deionized water. A native oxide layer,
which has a terminal thickness of ⬃0.6 nm, was then allowed to grow on each sample by exposure to ambient air for
at least 100 h.31 For our experiments this surface oxide is
ideal since it is stable in ambient but also thin enough to
permit the resistivity measurements to be made.
FIG. 1. Time dependent reflectivity change of Si on a 2-ps time
scale. Reflectivity is characterized by a coherent spike 共A兲, pulsewidth limited drop 共B兲, continuing slower drop 共C兲, and subsequent
rise 共D兲. Inset illustrates coherent-phonon contribution to reflectivity. Lower part of figure illustrates azimuthal-angle dependence of
coherent-spike amplitude.
lation of the reflectivity with a peak-to-peak amplitude in
⌬R/R of ⬃7⫻10⫺6 . This modulation is caused by coherent
excitation of the zone-center optical phonon and is discussed
in detail elsewhere.32
Figure 2 shows data obtained from three of the samples
with 120-ps scans. These data indicate the range of variation
in recovery of the reflectivity back towards its initial value
for the native-oxide terminated samples. Part 共b兲 of Fig. 2
shows the same data as in part 共a兲, with the ordinate expanded to more clearly illustrate the time dependence of the
recovery.
III. EXPERIMENTAL RESULTS
Reflectivity data obtained with 6.67 fs steps are shown in
Fig. 1. Data from all four samples are essentially identical to
the data shown in the figure. Initially, there is a small, sharp
increase in reflectivity 共labeled A兲 followed by a much larger
pulse-width limited decrease 共B兲. After this fast decrease the
reflectivity continues to decrease, but at a much slower rate
共C兲. The reflectivity then begins to recover approximately
500 fs after the initial excitation 共D兲. As is evident in comparing the data in the lower part of Fig. 1, the initial increase
in reflectivity 共A兲 is larger when the pump is polarized along
共010兲 compared to the 共110兲 direction.
As is emphasized in the inset in Fig. 1, data taken with the
pump polarized along 共110兲 exhibit a small sinusoidal modu-
IV. CONTRIBUTIONS TO SILICON REFLECTIVITY
CHANGES
In our measurements the 1.55 eV pump pulse excites electrons from states close to the valence band maximum at the
Brillouin zone center (⌫) to states near the conduction band
minimum, which is located at six equivalent k-space positions approximately 3/4 of the distance from ⌫ to X 关along
the 共001兲 and equivalent directions兴. Because the pump light
is linearly polarized, initial excitation into the six electron
pockets is anisotropic.33 This anisotropic distribution gives
rise to a degenerate four-wave mixing response that is responsible for the initial transient, part A of the reflectivity
curve shown in Fig. 1共a兲. As we show below, this nonlinear
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PHYSICAL REVIEW B 66, 165217 共2002兲
FEMTOSECOND PUMP-PROBE REFLECTIVITY STUDY . . .
We thus consider in detail free-carrier, state-filling, and
lattice-temperature contributions to the reflectivity.
In general, changes in reflectivity of a material can be
related to changes in the dielectric function. The preexcitation, room-temperature dielectric function of Si at 800 nm is
␧ˆ 0 ⫽13.656⫹i0.048. 29 However, because Re(␧ˆ 0 )ⰇIm(␧ˆ 0 )
and for our experimental conditions 兩 ⌬␧ˆ 兩 Ⰶ 兩 ␧ˆ 0 兩 and
Re(⌬␧ˆ )ⲏIm(⌬␧ˆ ), (⌬␧ˆ is the change in the dielectric function兲 we only need to consider Re(⌬␧ˆ ) in calculating or
modeling the reflectivity changes. In terms of the complex
index of refraction n̂⫽n⫹i ␬ ⫽(␧ˆ ) 1/2 this is equivalent to
only considering the contribution of ⌬n to the reflectivity
variations. Under these conditions the normalized change in
s-polarized reflectivity R s is related to ⌬n with very little
loss of accuracy (⬍0.1%) via
4n 0 cos共 ␪ 兲
⌬R s
⫽ 2
⌬n,
Rs
共 n 0 ⫺1 兲关 n 20 ⫺sin2 共 ␪ 兲兴 1/2
FIG. 2. Time dependent reflectivity of Si on a 120-ps time scale.
共a兲 The three curves illustrate the variation in measured reflectivity
for native-oxide covered Si共001兲. 共b兲 Measured 共dots兲 and calculated 共solid curves兲 reflectivity changes. The calculated curves correspond to values of S equal to 2.0 ⫻104 , 3.4 ⫻104 , and 4.6 ⫻ 10 4
cm s⫺1 .
optical response contains information on carrier momentum
relaxation in Si. The remaining variations in reflectivity are
linear-response induced, approximately sequentially due to
共1兲 free-carrier excitation, 共2兲 intraband carrier energy relaxation, and 共3兲 conduction-band to valence-band recombination. As we discuss below, these three processes are responsible for parts B, C, and D, respectively, of the reflectivity
curve in the upper part of Fig. 1. In the subsequent two
sections we detail the linear- and nonlinear-response contributions to the reflectivity changes. Our primary goal in these
two sections is to obtain a quantitative description of the
contributions to the reflectivity changes in order to construct
a time-dependent model that can be used to analyze the reflectivity data.
A. Linear-response contributions
There are two types of linear-response contributions to the
measured reflectivity changes: changes in the dielectric function associated with the presence of free-carriers and changes
in the dielectric function associated with interband transitions. The free-carrier contribution is commonly described
with the Drude model. The interband contribution can arise
from three separate effects: state filling, lattice temperature
changes, and band-gap renormalization. For our experimental conditions it appears band-gap renormalization changes
are insignificant 共see discussion at the end of this section兲.
共2兲
where ␪ is the angle of incidence and n 0 ⫽Re(␧ˆ 1/2
0 ). Because
the free-carrier 共FC兲, state-filling 共SF兲, and latticetemperature 共LT兲 contributions to ⌬n are all much smaller
than n we can write the total change in refractive index as
⌬n⫽⌬n FC⫹⌬n SF⫹⌬n LT .
Both the free-carrier and state-filling contributions to the
reflectivity depend upon the time dependence of the excited
carrier density N and carrier temperature T e . The carrier
density N first increases due to pump-pulse excitation and
then decreases via recombination. As is evident below from
the data analysis, at short times the recombination contribution to the time dependence of N is described very well by
exponential decay 共effective relaxation time ⬅ ␶ R ) to a nonzero background. That is, for an initial excitation density N in
the short-time time dependence of N can be expressed as
N 共 t 兲 ⫽N in
1⫹C R exp共 ⫺t/ ␶ R 兲
,
1⫹C R
共3兲
where the constant C R defines the nonzero background for
the decay.
Just after initial excitation the carriers each have an average excess energy of 0.30 eV 共above the band edge兲.34 As
discussed in detail in Sec. V. A, this excess energy becomes
thermalized among the carriers on a time scale that is significantly longer than the laser pulse width, so that a true carrier
temperature only develops on a time scale comparable to the
carrier-lattice equilibration time. However, for simplicity in
our analysis we assume that the carriers can be described by
an effective temperature. For Si the initial excess carrier energy of 0.3 eV corresponds to a hot-carrier temperature of
1985 K.35 The hot carriers equilibrate with the lattice with a
time constant ␶ el -ph—known as the electron-phonon energy
relaxation time—which is on the order of a few hundred
fs.8,9,14 The carriers and lattice equilibrate very close to RT
because of the relatively small number of carriers that are
excited. The electron-phonon energy relaxation time is only a
very weak function of excess carrier temperature;8,9,14 hence,
165217-3
PHYSICAL REVIEW B 66, 165217 共2002兲
A. J. SABBAH AND D. M. RIFFE
for an initial carrier temperature T in the time dependence of
the carrier temperature can be written as
T e 共 t 兲 ⫽ 共 T in⫺300 K兲 exp共 ⫺t/ ␶ el -ph 兲 ⫹300 K.
共4兲
Using the Drude model, the free carrier contribution ⌬n FC
to ⌬n can be expressed as6
⌬n FC共 N,T e 兲 ⫽⫺
2␲e2
N
* 共Te兲
n 0 ␻ m opt
2
,
共5兲
where ␻ is the 共angular兲 laser frequency, e is the electron
* (T e )⫽(1/m *e ⫹1/m h* ) ⫺1 is the carrier effeccharge, and m opt
tive optical mass. Here m e* and m h* are the electron and hole
conductivity effective masses, respectively. The T e depen* arises from band nonparabolicity.36 Recently
dence of m opt
we calculated the temperature dependences of m e* and m h* ,
* . 37 The calculation yields m opt
* (1985 K)
and thus m opt
* ⫽0.205 m e and m opt
* (300 K)⬅m 300
* ⫽0.156 m e (m e
⬅m 1985
is the free electron mass兲. With N⫽N 0 , Eq. 共5兲 yields
⌬n FC(N 0 ,1985 K)⫽⫺2.1⫻10⫺3 and ⌬n FC(N 0 ,300 K)⫽
⫺2.8⫻10⫺3 . Thus, the free carrier response initially produces a negative contribution to the reflectivity change, and
as the carriers cool the negative change increases in magnitude. At longer times, when recombination becomes important, the decrease in N, 关Eq. 共3兲兴, serves to decrease the freecarrier contribution to the reflectivity change.
* (T e ) at other temperatures are
Theoretical results for m opt
* (T e ) increases
summarized in Fig. 3, which shows that m opt
* (T e ) exponennearly linearly vs T e above RT. Therefore m opt
* with the time constant
tially decays to its RT value m 300
␶ el -ph . Putting together the expected time dependences of N
* , we obtain a time dependent free-carrier response
and m opt
function 共defined for t⬎0)
0
A FC共 t 兲 ⫽⫺A FC
⫻
1
* /m 300
* 兲 exp共 ⫺t/ ␶ el-ph兲
1⫹ 共 ⌬m opt
1⫹C R exp共 ⫺t/ ␶ R 兲
,
1⫹C R
共6兲
FIG. 3. Calculated carrier effective masses versus temperature.
共a兲 Conductivity effective masses of holes and electrons. 共b兲 Optical
effective mass. The dashed line is a linear fit to the calculation
between RT and 2000 K.
periment the noncolinearity and finite spot sizes of the pump
and probe beams introduce a small correction that results in
a slightly larger effective ␶ p . For our experimental geometry
we calculate a 6% correction factor, which results in an effective ␶ p ⫽25⫾1 fs.) Using this form for E(t), Eq. 共6兲 for
A FC(t), and the relationships ␶ p Ⰶ ␶ el-ph , ␶ p Ⰶ ␶ R , Eq. 共7兲 can
be expressed, to very good approximation, as
0
* ⫽m 1985
* ⫺m 300
* and A FC
is a positive constant.
where ⌬m opt
The free-carrier, time-dependent contribution to the measured reflectivity change is calculated from A FC(t) via
⌬R FC
共 ⌬t 兲 ⫽
Rs
冕
⬁
⫺⬁
⫻
冕
max
⌬R FC
⌬R FC
关 erf共 ⌬ t/ ␶ p 兲 ⫹ 1兴 /2
共 ⌬t 兲 ⫽
Rs
R s 1⫹ 共 ⌬m opt
* /m 300
* 兲 exp共 ⫺⌬t/ ␶ el-ph兲
2
dt 兩 E (p)
x 共 t⫺⌬t 兲 兩
t
⫺⬁
2
dt ⬘ A FC共 t⫺t ⬘ 兲 兩 E (e)
y 共 t ⬘ 兲兩 .
⫻
共7兲
(p)
Here E (e)
y (t) and E x (t) are the 共envelope兲 electric fields
associated with the excitation 共pump兲 and probe pulses, respectively, and ⌬t is the time delay between these two
pulses. In our experiment the excitation and probe electric
fields are well approximated by the Gaussian function E(t)
⫽E 0 exp关⫺(t/␶p)2兴, where ␶ p is related to the pulse intensity
FWHM ␶ FWHM via ␶ p ⫽ ␶ FWHM/2ln(2)⫽24⫾1 fs. 共In our ex-
1⫹C R exp共 ⫺⌬t/ ␶ R 兲
.
1⫹C R
共8兲
max
/R s is the Drude contribution to the change
The term ⌬R FC
in reflectivity for N⫽N 0 and T e ⫽300 K.
Pump-pulse excitation also modifies the interband dielectric function due to redistribution of electrons from the valence to conduction band: after excitation there are fewer
electrons that can be excited from the valence band and
fewer empty states in the conduction band that can be filled.
In general this state filling effect decreases the absorption
coefficient ␣ (⫽4 ␲ ␬ /␭) for photon energies above the band
165217-4
PHYSICAL REVIEW B 66, 165217 共2002兲
FEMTOSECOND PUMP-PROBE REFLECTIVITY STUDY . . .
gap. Since the excited electron and hole distributions depend
upon N and T e , the change in ␣ due to state filling also
depends on N and T e , i.e., ⌬ ␣ SF⫽⌬ ␣ SF(N,T e ). Via a
Kramers-Kronig relationship the change in the real part of
the index of refraction ⌬n SF(N,T e ) can be calculated from
the photon energy (E ph) dependence of ⌬ ␣ SF(N,T e ). 38 In
order to estimate ⌬ ␣ SF(N,T e ) and ⌬n SF(N,T e ), we assume
that ␣ ⫽K ␣ (E ph⫺E g ) 2 , which is theoretically expected for
indirect 共phonon assisted兲 absorption between parabolic
bands39 and is observed for Si for photon energies between
1.5 and 2.1 eV.29 Here K ␣ is a constant 共which we have
determined by fitting absorption coefficient data29兲, and E g
⫽1.12 eV is the indirect RT band gap of Si.
With this model we have calculated ⌬ ␣ SF(N,T e ) and
⌬n SF(N,T e ). The calculated value for ⌬ ␣ SF(N,T e ) at E ph
⫽1.55 eV is expected to be quite accurate since 1.55 eV is
in the range where ␣ ⫽K ␣ (E ph⫺E g ) 2 is valid. However, because the Kramers-Kronig calculation of ⌬n SF(N,T e ) depends upon values of ⌬ ␣ SF(N,T e ) at photon energies much
greater than E ph⫽1.55 eV where ␣ ⫽K ␣ (E ph⫺E g ) 2 no
longer holds, the calculated values of ⌬n SF(N,T e ) are only
semiquantitative in nature. For N⫽N 0 the model yields
⌬n SF(N 0 ,1985 K)⫽⫺5.0⫻10⫺4 and ⌬n SF(N 0 ,300 K)⫽
⫺2.4⫻10⫺4 . Although these values are only semiquantitative, we expect that the negative sign associated with
⌬n SF(N,T e ) and the increase in magnitude with temperature
to be robust results. Thus, state filling also produces an initially negative contribution to the reflectivity change. However, in contrast to the free-carrier contribution the negative
change becomes smaller in magnitude as the carriers cool to
RT. Comparing the Drude-model calculations of
⌬n FC(N 0 ,T e ) vs T e and these estimations for ⌬n SF(N 0 ,T e )
vs T e , we expect the 共negative兲 free-carrier change to dominate the 共positive兲 state-filling change as the carriers cool.
Thus the calculations indicate that the reflectivity should
continue to decrease as the carriers cool. This is in agreement
with the experimental observation of a continued decrease in
reflectivity up to a time delay of ⬃500 fs as shown in part C
of the reflectivity curve in Fig. 1.
In order to derive a response function due to state-filling
we note that our calculation of ⌬ ␣ SF(N,T e ) is also quite
linear as a function of T e between 300 and 2000 K. With the
assumption that ⌬n SF(N,T e ) is also linear as a function of
T e , we obtain a state-filling response function of the form
冋 冉
0
1⫹
A SF共 t 兲 ⫽⫺A SF
⌬n 1985
⫺1
⌬n 300
⫻exp共 ⫺t/ ␶ el-ph兲
册
冊
1⫹C R exp共 ⫺t/ ␶ R 兲
,
1⫹C R
共9兲
0
is a positive constant and ⌬n T ⬅⌬n SF(N 0 ,T).
where A SF
Here we have also assumed that the response function is
proportional to N, which is valid at nondegenerate excitation
densities (Nⱗ1020 cm⫺3 ). Analogous to Eq. 共8兲 for the freecarrier contribution to ⌬R s /R s , the state filling contribution
can be written as
冋 冉
max
⌬R SF
⌬R SF
erf共 ⌬t/ ␶ p 兲 ⫹1
⌬n 1985
1⫹
⫺1
共 ⌬t 兲 ⫽
Rs
Rs
2
⌬n 300
⫻exp共 ⫺⌬t/ ␶ el -ph 兲
册
冊
1⫹C R exp共 ⫺⌬t/ ␶ R 兲
.
1⫹C R
共10兲
The lattice-temperature contribution to the reflectivity
variations can be deduced from experimental measurements
of dn/dT L . We use dn/dT L ⫽3.4⫻10⫺4 K⫺1 , which we
have obtained by extrapolating measured values of dn/dT L
at wavelengths between 450 and 750 nm.40 We calculate the
rise in lattice temperature due to the excited electrons and
holes relaxing to their respective band edges to be 0.30
⫾0.02 K. Using
⌬n LT⫽
dn
⌬T L ,
dT L
共11兲
this rise in temperature produces ⌬n LT⫽1.0⫻10⫺4 . Thus, a
rise in lattice temperature serves to increase the reflectivity.
However, the lattice-temperature contribution is negligible
compared to the free-carrier and state-filling contributions, at
least before recombination becomes important. We thus neglect this contribution in analysis of the short-time reflectivity. At longer times when recombination dominates the dynamics and the band-gap energy is transferred back to the
lattice with each recombination event, the lattice-temperature
contribution to the reflectivity can become significant. Thus
in analyzing the 120-ps scans 共see Sec. V B兲 we include the
lattice-temperature contribution.
Any contribution to the reflectivity changes from bandgap renormalization is rather difficult to quantify. In principle, at high enough excitation densities carrier many-body
effects can effectively reduce the optical band gap, resulting
in a change in dielectric function that is equivalent to an
increase in the photon energy.41– 43 For Si this would produce
a positive change in n.29 However, this effect is significantly
different than the effects discussed above in that 共1兲 there is
expected to be a critical carrier density N cr 共slightly larger
than the Mott density兲 below which band-gap renormalization is absent,41 and 共2兲 for densities above N cr the change in
refractive index is expected to slowly vary with N, typically
as N 1/3 or N 1/4. 44 Using the expression of Bennett et al.,41 we
calculate N cr⫽6⫻1018 cm⫺3 , slightly above our excitation
density. Thus, the effect of band-gap renormalization on the
reflectivity may not be significant. Further, at excitation densities close to or significantly exceeding N cr , prior ultrafast
experiments on Si show no evidence of a band-gap renormalization contribution to the change in optical constants.6,9,10,17
We thus adopt the viewpoint that any band-gap renormalization contribution to the reflectivity is minimal and can thus
be neglected in modeling our measured reflectivity changes.
We note that in analyzing our experimental results nearsurface values of ⌬n can be used. This is because for our
experimental conditions the spatial variation of ⌬n is nearly
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PHYSICAL REVIEW B 66, 165217 共2002兲
A. J. SABBAH AND D. M. RIFFE
always much greater than the effective sampling depth of
␭/(4 ␲ 兩 n̂ 0 兩 )⫽17 nm. This is discussed in detail in the Appendix.
B. Coherent-transient contributions
The initial transient spike observed in our measurements
is similar to coherent transients previously observed using
other pump-probe experimental geometries that are also sensitive to degenerate four-wave mixing in semiconductors.
These geometries include transmission,45 two-beam
and
reflective
transient-grating
self-diffraction,15,46
diffraction.14,24
In our discussion of coherent transient effects we are
guided by the degenerate four-wave mixing theory of Smirl
and co-workers.47,48 There are two contributions to the
change in intensity in the 共reflected兲 probe-beam direction.
The first contribution arises from pump light that is diffracted from the polarization grating that is formed by the
overlap of the 共orthogonally polarized兲 pump and probe
beams at the sample surface. This polarization grating arises
from the anisotropic momentum distribution of the initially
excited carriers. The anisotropy decays via carrier-carrier and
carrier-phonon scattering on a timescale that is significantly
less than 100 fs.15,24 Because this momentum randomization
is significantly faster than the other relaxation processes 共energy relaxation, recombination, and diffusion兲 that govern
the carrier dynamics, the response function associated with
scattering from the polarization grating can be written, assuming exponential relaxation, as47
(3) (3)
A (3)
xy yx 共 t 兲 ⫽A 0 Y xy yx exp共 ⫺t/ ␶ ␪ 兲 .
共12兲
Here ␶ ␪ is the relaxation time associated with momentum
(3)
randomization, A (3)
0 is a constant, and Y i jkl is a dimensionless tensor with the symmetry of the third-order susceptibility ␹ (3)
i jkl . The contribution to the change in reflectivity from
this response function is given by
⌬R PG
共 ⌬t 兲 ⫽
Rs
冕
⬁
⫺⬁
⫻
冕
mation through its time dependence, but only produces a
coherent spike centered at ⌬t⫽0 with a width given by the
convolution of the pump and probe pulses.
The second degenerate four-wave mixing contribution to
the change in reflectivity arises from the probe beam interacting with the anisotropic component of the pump induced
carrier distribution. This contribution is governed by the response function47
(3)
(3)
(1) (1)
A (3)
xxy y 共 t 兲 ⫽A 0 共 Y xxy y ⫺Y xx Y y y 兲 exp共 ⫺t/ ␶ ␪ 兲 .
Similar to the third-order susceptibility symmetry terms,
has the symmetry of the linear susceptibility ␹ (1)
Y (1)
ij
ij ,
which for cubic Si is proportional to ␦ i j , the Kroneker delta.
The contribution of this response function to the reflectivity
is obtained using Eq. 共7兲 with A FC(t) replaced by A (3)
xxy y (t).
Again, the integrals can be analytically solved to yield the
anisotropic-distribution contribution to the reflectivity
change
冋 冉
0
⌬R AD
⌬R AD
⌬t
␶p
erf
⫺
共 ⌬t 兲 ⫽
Rs
Rs
␶p
2␶␪
⫺⬁
共13兲
For Gaussian laser pulses and the exponential response
function of Eq. 共12兲, the integrals in Eq. 共13兲 can be solved
exactly to yield the polarization-grating contribution to the
reflectivity
冉 冊
⫹ 1 exp
⫺⌬t
.
␶␪
共16兲
1
(3)
Y (3)
xy yx 共 ␾ 兲 ⫽Y 1221⫹ 关 1⫺cos共 4 ␾ 兲兴
4
(e)
(p)
dt ⬘ A (3)
xy yx 共 t⫺t ⬘ 兲 E y * 共 t ⬘ 兲 E x 共 t ⬘ ⫺⌬t 兲 .
0
⌬R PG
⫺⌬t 2
⌬R PG
exp
.
共 ⌬t 兲 ⫽
Rs
Rs
␶ 2p
冊 册 冉 冊
As shown below, for our experimental conditions ␶ p
⬇ ␶ ␪ . Even so, Eq. 共16兲 produces a peak with a significant
tail that is described by the exponential decay exp(⫺⌬t/␶␪).
This contribution thus provides information on carrier momentum relaxation.
In a cubic semiconductor the components of Y (3)
i jkl depend
upon the orientation of the sample. In our experiment a rotation of the sample about its azimuth can distinguish between the linear-optical effects discussed in Sec. IV A and
the nonlinear-optical effects discussed here. For example, if
(3)
to be Y (3)
we define Y 1221
xy yx when the pump and probe polarizations are oriented along the cubic axis directions of the
sample, then upon rotating the crystal through an angle ␾
15
about its azimuth Y (3)
xy yx transforms as
(e)
dt E (p)
x * 共 t⫺⌬t 兲 E y 共 t 兲
t
共15兲
(3)
(3)
(3)
(3)
⫻ 共 Y 1111
⫺Y 1122
⫺Y 1212
⫺Y 1221
兲.
共17兲
The term Y (3)
xxy y transforms similarly. Thus, for the two
sample orientations in our experiment ( ␾ ⫽0 and ␾ ⫽ ␲ /4)
we obtain the largest possible differences in the probe intensity associated with ⌬R PG and ⌬R AD . This difference is
clearly observed in the initial spike in the data shown in the
lower part of Fig. 1.
共14兲
V. DATA ANALYSIS AND DISCUSSION
A. Momentum and energy relaxation
0
The parameter ⌬R PG
/R s is a function of ␶ p and ␶ ␪ .
(3)
0
Through the term Y xy yx , ⌬R PG
/R s also depends on the orientation of the sample. However, the time-delay (⌬t) dependence of Eq. 共14兲 is solely determined by the pulse-width
parameter ␶ p . Thus, this term contains no dynamical infor-
Using the sum of Eqs. 共8兲, 共10兲, 共14兲, and 共16兲, along with
a model equation for the coherent phonon excitation and
decay,32 we have fit the short-time-scale data with a leastsquares analysis. In total there are 16 parameters used to
analyze the data. There are 10 associated with carrier dynam-
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PHYSICAL REVIEW B 66, 165217 共2002兲
FEMTOSECOND PUMP-PROBE REFLECTIVITY STUDY . . .
FIG. 4. Least-squares analysis of Si time dependent reflectivity.
共a兲 Data, overall fit, and components 关free carrier 共FC兲, state filling
共SF兲, polarization grating 共PG兲, and anisotropic distribution 共AD兲兴
are shown. The individual components are shifted by 0.25 ps. 共b兲
Residuals of the fit shown in 共a兲. 共c兲 Residuals of a fit to the data
with only the FC, SF, and PG components.
max
max
0
0
ics (⌬R FC
/R s , ⌬R SF
/R s , ⌬R PD
/R s , ⌬R AD
/R s , ␶ ␪ , ␶ el-ph ,
* /m 300
* , n 1985/n 300 , and C R ), 2 associated with the
␶ R , ⌬m opt
laser pulses ( ␶ p and the zero-time delay兲, and four others
associated with the coherent-phonon response.32 However,
because the relaxation of the free-carrier and state-filling responses are characterized by the same relaxation processes,
max
max
* /m 300
* , and
/R s , ⌬R SF
/R s , ⌬m opt
the parameters ⌬R FC
n 1985/n 300 are highly correlated and cannot all be uniquely
determined from the data. We have thus used our calculations to place two constraints on the fitting: we con* /m 300
* to the theoretical value of 0.31
strain ⌬m opt
max
max
and ⌬R SF
/⌬R FC
to our calculated value of
⌬n SF(N 0 ,300 K)/⌬n FC(N 0 ,300 K)⫽0.086.
A typical least-squares fit with these two constraints is
shown in Fig. 4共a兲. Along with the data 共points兲 and overall
fit 共solid line through the data兲, we also display the four
individual contributions to the reflectivity 共shifted by 0.25 ps
for clarity兲. The statistical nature of the residuals in Fig. 4共b兲
indicates that the model provides an excellent description of
the data. From analysis of all of our data we obtain ␶ ␪ ⫽32
⫾5 fs, ␶ el-ph⫽260⫾30 fs, n 1985/n 300⫽1.7⫾0.3, and ␶ p
⫽26⫾1 fs. This extracted value of ␶ p is in excellent agreement with the autocorrelator derived value of 25⫾1 fs. The
value of n 1985/n 300 is in very good agreement with our theoretical estimate of 2.1, which suggests that the relative contributions from the free-carrier and state-filling responses are
reasonable
well
described
by
the
constraint
⌬n SF(N 0 ,300 K)/⌬n FC(N 0 ,300 K)⫽0.086. However, we
emphasize that the values of ␶ ␪ and ␶ el-ph are largely insensitive to the two imposed constraints. For example, setting
the state-filling contribution to zero and using only the freecarrier contribution to describe the linear response results in
␶ el-ph⫽240⫾30 fs and no change in ␶ ␪ . Although the contribution to the overall reflectivity from ⌬R AD/R s is rather
small, its inclusion in the analysis is necessary in order to
quantitatively describe the reflectivity curve. This is illustrated by the structure present in the residuals if a fit is per0
/R s constrained to zero, as shown in Fig.
formed with ⌬R AD
4共c兲.
Our value of 32⫾5 fs for ␶ ␪ is, to our knowledge, the first
ultrafast experimental determination of this parameter. Because of lower time resolution, previous investigations have
only suggested the time scale for this parameter. The degenerate four-wave mixing experiments of Buhleier et al. with
2.0 eV, 100 fs pulses indicated momentum relaxation on a
time scale of 10 fs.15 The transient grating experiments of
Sjodin et al. with 1.55 eV, ⬃100 fs pulses also showed that
the momentum reorientation time is significantly less that
100 fs.24
Theoretical calculations suggest that carrier-phonon scattering slightly dominates the carrier-carrier scattering contribution to ␶ ␪ . Theoretical electron-phonon and hole-phonon
scattering times have been calculated by a number of researchers that use Monte Carlo simulations to model hotcarrier transport. For an excess carrier energy of 0.3 eV
carrier-phonon scattering times range between 20 and 60
fs.2,3,26 Carrier-carrier scattering rates for carrier densities
and temperatures appropriate to our experiment have been
calculated in a series of papers by Combescot et al. and
Sernelius.25,49 From their theoretical expressions we calculate electron-electron and electron-hole scattering times of
155 and 170 fs, respectively, for our initial excitation density
and excess carrier energy.50 Combining these scattering rates
共assuming that they are independent兲 yields an overall
carrier-carrier scattering time of 80 fs. Combining this rate
with the range of theoretical carrier-phonon rates yields an
overall relaxation time between 16 and 35 fs, entirely consistent with our measured value of ␶ ␪ ⫽32⫾5 fs.
The carrier energy-relaxation time ␶ el-ph is an important
parameter in Monte Carlo modeling of semiconductor
devices.1– 4 Our result, ␶ el-ph⫽260⫾30 fs, is most directly
comparable to the result of Sjodin et al., who obtain an average value of 240⫾30 fs for excitation densities less than
⬃1020 cm⫺1 . 14 Our result also compares favorably with a
dc-transport inferred value of 270 fs.51 For comparison with
Monte Carlo derived values of ␶ el-ph we have plotted both
experimental and theoretical values of ␶ el-ph as a function of
excess carrier energy in Fig. 5. Experimental data from three
previous studies,8,9,14 in addition to the present study, are
represented by the solid symbols and solid line 共the solid line
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PHYSICAL REVIEW B 66, 165217 共2002兲
A. J. SABBAH AND D. M. RIFFE
bulk scattering processes. It is therefore not surprising that
photoemission inferred values for carrier-carrier and carrierphonon scattering processes are somewhat different than
those from our bulk-sensitive measurements. Furthermore,
even though the two photoemission studies were of nominally the same Si(001)2⫻1 surface, the observed differences can be attributed to differences in surface defect density on the measured Si surfaces.23
B. Carrier recombination
FIG. 5. Comparison of experimental and theoretical values of
␶ el -ph . Calculated values are for electron cooling 共dashed curve兲
and hole cooling 共open symbols and thin solid curves兲. Experimental values are represented by solid symbols and the thick solid line.
is a linear fit to the data of Sjodin et al.14兲. In determining the
excess carrier energy for the experimental values of ␶ el-ph
both linear29 and two-photon absorption9,42 have been included. From the study of Sjodin et al. we have only used
data at carrier densities less than ⬃2⫻1020 cm⫺3 , above
which intracarrier screening affects the carrier-phonon
interaction.52 Monte Carlo calculations of both the hole energy decay time 共open symbols with dotted lines兲 and electron energy decay time 共dashed line兲 are also shown.4,53 The
comparison between experimental and theoretical values of
␶ el -ph indicates that the UTMC and Damocles Monte Carlo
codes4 produce values for ␶ el-ph that are closest to the experimental data. We also note that a very recent measurement
indicates ␶ el-ph⫽240 fs for nanocrystalline Si.54
There have been several ultrafast photoemission studies
of the Si(001)2⫻1 surface.20–23 In principle, these measurements are able to follow both intraelectron thermalization
and hot-electron cooling by measuring the time dependence
of the electron distribution function. Indeed, using 2.0 eV,
⬃120 fs pump pulses Goldman and Prybyla observed thermalized electron distributions characterized by temperatures
of 1200 and 800 K at 120 and 180 fs delay times, respectively. Their results indicate that electron-electron thermalization 共and hence electron-electron scattering兲 is significantly faster than ⬃100 fs and that ␶ el -ph is ⬃90 fs. 21 In
contrast, the photoemission data of Jeong and Bokor, obtained with 1.55 eV, 150 fs pump pulses, are consistent with
an intraelectron thermalization time of a few hundred fs and
a value for ␶ el-ph of 400 fs.23 The discrepancy among results
from the two photoemission studies and the present study
can be explained by the surface-sensitive nature of the photoemission process,20,23 which for Si(001)2⫻1 is dominated
by electrons photoemitted from defect-induced surface
states.23 Thus, in the photoemission measurements surfacestate dynamics obscure bulk-carrier dynamics so that measured relaxation times cannot be directly associated with
The recovery of the reflectivity signal shown in 120 ps
scans of Fig. 2 is predominantly due to the reduction in excited carrier density N as electrons and holes recombine
across the Si band gap. Both the free-carrier 关Eq. 共5兲兴 and
state-filling contributions to the reflectivity diminish with recombination. For our samples the recombination process is
dominated by Shockley-Reed-Hall 共SRH兲 recombination involving surface band-gap states.11 In the SRH recombination
process the electron-hole energy (⬃1.2 eV per electron-hole
pair兲 is absorbed by the lattice via multiphonon emission.
This energy return to the lattice can produce a non-negligible
contribution to the reflectivity change from the associated
increase in lattice temperature T L 关Eq. 共11兲兴 near the sample
surface.11
In order to describe the time dependence of N and T L
appropriate to the time scale of Fig. 2 we use a model that
includes diffusion of both carrier density N and lattice temperature T L . 10,11,17 The diffusion equations for N and T L are
coupled through boundary conditions that depend upon the
surface recombination velocity S, which describes the rate of
SRH recombination via surface states. The recombination
velocity S, the only free parameter in the diffusion model, is
proportional to effective surface-state trap density n S and
cross section ␴ , and carrier thermal velocity v th
⬇107 cm s⫺1 ; S⬇ ␴ v thn S . 55 If we assume that ␴ ⬇1/n A ,
where n A ⫽6.8⫻1014 cm⫺2 is the surface-atom density on
the Si共001兲 surface, then S⬇ v thn s /n A . Because the electrons
and lattice can be considered to be in thermal equilibrium,
the relative contributions to ⌬R s /R s from N and T L can be
determined from ⌬n FC(N,T L ), ⌬n SF(N,T L ), and ⌬n LT(T L ),
as described in Sec. IV A.
Using this model we have numerically solved the diffusion equation for the time dependence of near-surface values
of N and T L . In Fig. 6 we show calculated time-dependent
reflectivity changes that illustrate the separate reflectivity
contributions from ⌬N and ⌬T L for S⫽3⫻104 , S⫽3
⫻105 , and S⫽3⫻106 cm s⫺1 . Note that for a large enough
recombination velocity the T L contribution can dominate the
N contribution and cause the reflectivity to initially recover
past its initial value. We have previously observed this behavior on other Si surfaces.11
In Fig. 2共b兲 we compare the experimental reflectivity
curves with calculated reflectivity curves. In each comparison S has been adjusted to produce the best overall match to
the experimental data. Values of S used in the three calculated curves in Fig. 2共b兲 are 2.0⫻104 , 3.4⫻104 , and 4.6
⫻104 cm s⫺1 . Whereas the decrease in N on the few-ps time
scale of Figs. 1 and 4 can be described by an exponential
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PHYSICAL REVIEW B 66, 165217 共2002兲
FEMTOSECOND PUMP-PROBE REFLECTIVITY STUDY . . .
VI. SUMMARY
We have studied bulk carrier dynamics near native-oxide
covered Si共001兲 using ultrafast pump-probe reflectivity .
From the measurements we have determined the carrier momentum and energy relaxation time scales. The momentum
relaxation time ␶ ␪ ⫽32⫾5 fs is the first measurement of this
parameter in Si. Both carrier-phonon and carrier-carrier scattering appear to contribute to momentum relaxation. The carrier energy relaxation time ␶ el-ph⫽260⫾30 fs agrees well
with previous experimental determination of this parameter.
The experimental values of ␶ el-ph agree reasonably well with
recent Monte Carlo derived values. Finally, we have extracted a value for the surface recombination velocity of
native-oxide terminated Si共001兲, which is directly related to
the density of traps that participate in surface recombination.
APPENDIX
In this appendix we discuss the reflectivity from an inhomogeneously excited solid, such as occurs in the experiments
discussed here. In particular, we study the reflectivity of a
solid with a dielectric function that varies exponentially with
depth z into the solid as
␧ˆ 共 z 兲 ⫽␧ˆ 0 ⫺ ␩ˆ exp共 ⫺z/ ␦ 兲 .
FIG. 6. Calculated time-dependent reflectivity curves for Si
共solid lines兲 using carrier and temperature diffusion model 共see
text兲. Carrier 共free carrier and state filling兲 and lattice-temperature
contributions are shown as long-dashed and short-dashed curves,
respectively. 共a兲 S ⫽ 3 ⫻ 10 4 cm s⫺1 . 共b兲 S ⫽ 3 ⫻105 cm s⫺1 . 共c兲
S ⫽ 3 ⫻106 cm s⫺1 .
decay, the data of Fig. 2 and the results of the diffusionmodel calculation show that on a 120-ps time scale the recovery in reflectivity is relatively faster at short times and
slower at long times than a simple exponential decay. Additionally, the comparison in Fig. 2共b兲 between the calculations
and data shows that the measured short-time change is relatively faster than short-time change calculated in the model.
This is likely due to two effects.19,23 First, the recombination
velocity is expected to have some time dependence due to
transient occupation effects of the surface states.20 Additionally, surface-charge induced band bending is expected to
drive either electrons or holes away from the surface, decreasing the effective recombination velocity as a function of
time. Neither of these effects are included in the diffusion
model. For all of our native-oxide covered samples we extract a value for S of 共3 ⫾ 1兲 ⫻104 cm s⫺1 . Using S
⬇ v th n s /n A , this value of S corresponds to a surface-state
trap density of (0.3⫾0.1)⫻10⫺2 monolayer for native-oxide
terminated Si共001兲.
共A1兲
Here ␦ describes the decay length for relaxation of the
dielectric function to the bulk dielectric ␧ 0 and ⫺ ␩ˆ is the
change in ␧ˆ at the surface 共so that ␧ˆ 0 ⫺ ␩ˆ is the dielectric
function near the surface兲. The reflectivity for the two limits
of large and small ␦ are fairly easy to surmise. If ␦ is much
smaller than some appropriate length scale 共which as we
show below is related to the wavelength of the probe radiation inside the solid兲 so that the changes in optical constants
are confined to only a very small spatial region near the
surface, then we expect the reflectivity to be characterized by
the background dielectric function ␧ 0 . Conversely, if ␦ is
large, then we expect the reflectivity to be characterized by
␧ 0 ⫺ ␩ , the dielectric function near the surface. We show
here that the transition in reflectivity between these two limits occurs for ␦ ⫽d obs , where d obs⫽␭/(4 ␲ 兩 n̂ 0 兩 ). The parameter d obs is known as the effective observation depth of the
probe radiation. This length can often be much smaller than
the penetration depth of the probe radiation ␦ probe
⫽␭/(4 ␲ ␬ 0 ). However, as we show below, it is the ratio
␦ /d obs that defines the transition between the two reflectivity
extremes rather than the ratio ␦ / ␦ probe . The normal-incidence
reflectivity has previously been discussed in the literature:
Vakhnenko and Strizhevskii has solved this problem
exactly.56 However, the exact solution involves nonintegralorder Bessel functions. Here we outline a perturbation-theory
solution that provides much simpler expressions for calculating the reflectivity.
Assuming linear polarization, normal incidence, and harmonic time dependence the wave equation for the electric
field E(z) inside the solid can be written as
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PHYSICAL REVIEW B 66, 165217 共2002兲
A. J. SABBAH AND D. M. RIFFE
d 2E共 z 兲
dz 2
⫹k 2 ␧ˆ 共 z 兲 E 共 z 兲 ⫽0,
共A2兲
where k⫽ ␻ /c and ␧ˆ (z) is given by Eq. 共A1兲. The solution
for the electric field is found using perturbation theory,57
where the small parameter in the problem is ␩ˆ /␧ˆ 0 . 共For our
experiments 兩 ␩ˆ /␧ˆ 0 兩 ⬇ 0.02.兲 The solution to Eq. 共A2兲 to first
order in ␩ˆ /␧ˆ 0 is
再
E 共 z 兲 ⫽E 0 exp共 ikn̂ 0 z 兲 1⫹
␩ˆ k 2 ␦ 2
1⫺i ␦ /d̂ obs
冎
exp共 ⫺z/ ␦ 兲 ,
共A3兲
where d̂ obs⫽1/(4 ␲ n̂ 0 ). The normal-incidence reflectivity R
can be expressed as
R⫽
冏 冏
1⫺ẑ s
2
1⫹ẑ s
共A4兲
,
where
FIG. 7. Normalized Si reflectivity versus ␦ /d obs . Solid curve
corresponds to ␧ˆ 0 ⫽ 13.656 ⫹ i 0.048 and ␩ˆ ⫽ 0.02. Upper dashed
line corresponds to a uniform solid with ␧ˆ ⫽␧ˆ 0 . Lower dashed
curve corresponds to a uniform solid with ␧ˆ ⫽␧ˆ 0 ⫺ ␩ˆ .
where ẑ s0 is the surface impedance for ␩ˆ ⫽0. Equation 共A6兲
has the expected limits for small and large ␦ . For ␦ /d obs
Ⰶ1, ẑ s ⬇ẑ s0 , which is equivalent to the surface impedance
for a uniform material with ␧ˆ ⫽␧ˆ 0 , while for ␦ /d obsⰇ1, ẑ s
⬇ẑ s0 关 1⫹ ␩ˆ /(2␧ˆ 0 ) 兴 , which is equivalent to the surface impedance for a uniform material with ␧ˆ ⫽␧ˆ 0 ⫺ ␩ˆ .
Using Eq. 共A6兲 in Eq. 共A4兲 we have calculated the reflec-
tivity R 共normalized by the reflectivity R 0 for ␩ˆ ⫽0) as a
function of ␦ for conditions appropriate to the change in ␧ˆ in
our experiment: we use ␧ˆ 0 ⫽13.656⫹i0.048 and ␩ˆ ⫽0.02.
The results are plotted in Fig. 7. As can be seen in the graph,
for small ␦ /d obs the reflectivity approaches R 0 , while for
large ␦ /d obs the reflectivity approaches a constant value,
which is appropriate to ␧ˆ ⫽␧ˆ 0 ⫺ ␩ˆ . Note that for ␦ /d obs ⫽ 1
the reflectivity is exactly half way between the reflectivity
for the small and large ␦ /d obs limits. For the initial excitation
conditions in our experiment ␦ /d obs⫽570. Hence, initially
the probe-pulse reflectivity is characterized by the nearsurface dielectric function. At longer times, carrier and temperature diffusion produce a rather complicated
z-dependence to ␧ˆ . 11 However, after approximately 1.5 ps
the length scale for the change in ␧ˆ near the surface is greater
than 5 d obs . Thus, in modeling the 120-ps reflectivity
changes we also only need to consider the near-surface dielectric function.
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E
ẑ s ⫽ik
dE/dz
冏
共A5兲
z⫽0
is the normalized surface impedance. Substituting Eq. 共A3兲
into Eq. 共A5兲 and keeping terms up to linear order in ␩ˆ /␧ˆ 0
yields
冉
ẑ s ⫽ẑ s0 1⫹
1
␩ˆ
1
2␧ˆ o 1⫹id̂ obs/ ␦
冊
,
共A6兲
8
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